What is a good example of a nilpotent algebra?

An example of a nilpotent algebra is mathfrak{t}_n(mathbb{C}), the complex-valued, strictly upper triangular matrices. The Lie bracket here is the standard commutator, [x,y]=xy-yx. Now clearly repeated applications of the commutator to upper triangular matrices will result in the matrix ‘level’ (the sub-diagonal above the main diagonal) decreasing, and thus the lower central series will terminate.

section{Killing Form}

The Killing form is a bilinear, symmetric form on mathfrak{g} defined as a function k: mathfrak{g} times mathfrak{g} right arrow mathbb{C} such that

[k(x,y)=Tr(ad_x circ ad_y)]

This linear function produces ntimes n matrices when written on some basis, for any n-dimensional Lie algebra. Another important fact about k is that it is invariant begin{defn} If we have a bilinear form B such that, B([x,y],z)=B(x,[y,z]) for all x,y,z in mathfrak{g} then we call B invariant

end{defn} We will explicitly calculate the Killing form for mathfrak{sl}_2(mathbb{C}).

We begin by using the multiplication table used at the end of chapter 1, noting that:

[[e,f]=h, [h,e]=2e, [h,f]=-2f]

So by the properties of the Lie bracket we can see that:

[[e,e]=0,[f,f]=0,[h,h]=0,[f,e]=-h,[e,h]=-2x]

With respect to this basis (e,h,f) we can create matrices of the adjoint operators:

[ad_e=begin{pmatrix} 0 -2 0  0 0; 1 0 0 0end{pmatrix},ad_h=begin{pmatrix} 2  0 0 0 0 0 0 0 -2end{pmatrix}, ad_f=begin{pmatrix} 0  0 0 -1 0 0 0 2 0end{pmatrix}]

From these matrices we can calculate the Killing forms, so

[k(e,e)=0, k(h,h)=8, k(f,f)=0, k(e,f)=k(f,e)=4, k(e,h)=k(y,h)=0]